Final answer:
Solving the provided equation for the current speed using algebra and the total time for a round trip of the motorboat leads us to find that the river's current speed is 3 miles per hour.
Step-by-step explanation:
The student is asked to determine the speed of the current of a river given specific conditions during a round trip of a motorboat. The boat maintains a speed of 17 miles per hour relative to the water and the total distance covered upstream and downstream is 52 miles each way, taking a total of 8.5 hours. We let c be the current's speed in miles per hour.
To solve for c, we can use the formula for time:
time = distance / speed. The motorboat's effective speed going upstream is (17 - c) mph and downstream is (17 + c) mph. The total time is the sum of the upstream time and the downstream time:
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- Upstream: 52 / (17 - c)
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- Downstream: 52 / (17 + c)
The equation representing the total time spent on the trip is:
(52 / (17 - c)) + (52 / (17 + c)) = 8.5
This can be solved for c by finding a common denominator, resulting in a quadratic equation, and then using algebraic techniques to solve for c.
We simplify the equation to find that the current's speed c is 3 miles per hour.